Tropical modes of variability in Earth's climate system, such as the Madden-Julian Oscillation (MJO) and the El Niño-Southern Oscillation (ENSO), have global relevance for climate prediction and projection over various timescales. Yet they remain challenging to represent in state-of-the-art climate models (e.g., Chen et al., 2017;Hung et al., 2013). Over time, model development has generally led to encouraging improvements in their representation (Ahn et al., 2020;G. Wang et al., 2015). However, the complexity of contemporary, full-fledged Earth system models often obscures the sources of improvement (or lack thereof) regarding various aspects or processes (see, e.g., discussion in Bayr et al., 2019;Klingaman & Demott, 2020;Planton et al., 2021). Studying these modes of variability in an idealized framework can facilitate the understanding of their behavior in more complex models. For example, idealized models have been used to study the role of the Pacific mean state-including the zonal gradients in the atmosphere and the ocean-in setting the dynamics of both MJO (e.g.,